The sampling regions for the real-time symmetry transform are circular
concentric rings of varying size and thickness (see
Figure ). The use of circular rings is ideal for detecting
rotational invariance. However, we wish to develop an operator which will
detects blobs of a certain form. This requires specific annular sampling
regions to detect the contours of interest in the image. In other words, the
sampling region should approximate the shapes we are searching for in the
image. We proceed by defining a parameterized mathematical model of the
templates to approximate facial contours. Figure depicts
the geometry of this so-called semi-elliptical model. It is a superset of the
class of elliptical annuli which is itself a superset of the class of circular
annuli. Such a model lends itself quite well to detecting human face and head
contours which are not well-approximated by simple ellipses or circles. The
model can be viewed as two half ellipses spliced together along a shared axis
of length b. The model has the following parameters:

By varying the parameters of the model, we can generate the desired blob
templates. The orientations of the normals at the boundary of the template
are also pre-computed by a template generation process. Note that the
orientation of the normals varies not only as a function of the angular
position on the sampling region but also as a function of the radius from the
center. The center of the model is a well defined point: it is the
intersection of the shared elliptical axis and the meeting point of the other
two axes (a and c).

The equations for an for an ellipse's boundary and its contour's normals are used
to construct the templates [43]:

(2.6)

(2.7)

a

length of major axis (
a1<a<a2)

b

length of minor axis (
b1<b<b2)

angle of ellipse minor axis with respect to vertical

r

radius

angle between radius vector and boundary normal

Figure depicts a typical template consisting of a
semi-circular annulus on top of a semi-elliptical annulus. The orientation
values of the normals of the template are represented by intensity. Note the
similarity of this template to the shape of a human head or face.